# Divisibility Tests Unified: Stacking the Trimmings for Sums

## TL;DR

This paper unifies various divisibility tests, including trimming and summing types, by deriving the most effective summing tests from a generalized trimming test, using simple properties and the concept of stacking.

## Contribution

It presents a unified framework for divisibility tests by deriving summing tests from a generalized trimming test, simplifying understanding and application.

## Key findings

- Derived Khare's summing tests from Zbikowski's trimming test
- Showed binomial tests can be obtained from an adapted Zbikowski's test
- Introduced the concept of stacking to explain divisibility decision processes

## Abstract

Divisibility tests are algorithms that can quickly decide if one integer is divisible by another. There are many tests but most are either of the trimming or summing variety. Our goals are to present Zbikowski's family of trimming tests as one test and to unify the trimming and summing tests. We do the latter by showing, first, that the most effective summing tests, due to Khare, can be derived directly from the Zbikowski's test and, second, that the best known summing tests - the binomial tests - can be derived from an adapted form of Zbikowski's tests. We introduce the notion of stacking, the claim that a six year old would always choose 10 pennies over a dime, and use only basic divisibility properties to achieve our goals.